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Septischismic family: Difference between revisions

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{{Main| Septischismic }}
{{Main| Septischismic }}


The head of this family is septischismic (sometimes ''garischismic''), which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.  
The head of this family is septischismic (formerly ''garischismic''), which is generated by a [[3/2|perfect fifth]] and an independent generator for [[5/4]]. Two [[Pythagorean apotome]]s i.e. 14 fifths octave-reduced make a [[8/7|septimal major second (8/7)]]. Equivalently stated, the [[7/4|harmonic seventh (7/4)]] is found at the double-diminished octave (C–C𝄫), or the minor seventh minus a generic comma step which stands in for both the Pythagorean comma and the septimal comma.  


Septischismic can be easily notated with [[chain-of-fifths notation]] with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].  
Septischismic can be easily notated with [[chain-of-fifths notation]] with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step which stands in for the [[schisma]] and the [[aberschisma]].  
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Undecimal septischismic, a.k.a. cassaschismic, maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an [[expansion]] of the [[2.3.7.11 subgroup|2.3.7.11-subgroup]] version of [[gary]]. It is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.  
Undecimal septischismic, a.k.a. cassaschismic, maps [[prime interval|prime]] [[11/1|11]] to +23 perfect fifths, so it is an [[expansion]] of the [[2.3.7.11 subgroup|2.3.7.11-subgroup]] version of [[gary]]. It is naturally a no-17 19-limit temperament, where the undevicesimal schisma of [[513/512]] is also added to the generic aberschisma step.  


Septischismic has two reasonable extensions to add 17, both of which coalesce in the structurally very similar temperament [[newt]]. Septinoellismic finds it a tad unintuitively as two aberschismas up and 4 commas down from a major second (^^4sM2) yet supported by 94edo, and septisperasmic finds it 4 aberschismas up from a minor second (4^m2), which can be deemed the more intuitive mapping, albeit it is not supported by any cassandra tuning except for 12edo which notably has a very well tuned 17/16.
Septischismic has two reasonable extensions to add 17, which coalesce in the newt extension 270 & 311. Septinoellismic finds it a tad unintuitively as two aberschismas up and 4 commas down from a major second (^^4sM2) yet supported by 94edo, and septisperasmic finds it 4 aberschismas up from a minor second (4^m2), which can be deemed the more intuitive mapping, albeit it is not supported by any cassandra tuning except for 12edo, which notably has a very well tuned 17/16.
 
Septischismic has two reasonable extensions to add 29 and 31 into the system, that like before, coalesce in the newt extension 270 & 311 too. Septiplanar finds them in the chain of fifths, without need of third generators, thus they in the same plane to the rank-2 chain of fifths, mapped respectively to a major seventh – three commas (3sM7) and a perfect octave minus two commas (hP8). Because this extension is reached only through the septischismic chain of fifths, it is also available in [[gary]].


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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Comma list: 2080/2079, 4096/4095, 19712/19683
Comma list: 2080/2079, 4096/4095, 19712/19683


Mapping: {{mapping| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}
{{Mapping|legend=0| 1 0 0 25 -33 -13 | 0 1 0 -14 23 12 | 0 0 1 0 0 -1 }}


Optimal tunings:  
Optimal tunings:  
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Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 | 0 1 0 -14 23 12 5 | 0 0 1 0 0 -1 1 }}


Optimal tunings:  
Optimal tunings:  
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===== Septiplanar =====
===== Septiplanar =====
Septischismic has two reasonable extensions to add 29 and 31 into the system, that like before, coalesce in newt. Septiplanar finds them in the chain of fifths, without need of third generators, thus they in the same plane to the rank-2 chain of fifths, mapped respectively to a major seventh – three commas (3sM7) and a perfect octave minus two commas (hP8). Because this extension is reached only through the septischismic chain of fifths, it is also available in [[gary]].
Subgroup: 2.3.5.7.11.13.19.29
Subgroup: 2.3.5.7.11.13.19.29


Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 23 | 0 1 0 -14 23 12 5 -31 | 0 0 1 0 0 -1 1 0 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 23 | 0 1 0 -14 23 12 5 -31 | 0 0 1 0 0 -1 1 0 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.948{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.379{{c}}
* WE: ~2 = 1199.9476{{c}}, ~3/2 = 702.2134{{c}}, ~5/4 = 386.3794{{c}}
: error map: {{val| -0.052 +0.206 -0.039 -0.389 +0.113 -0.190 -0.119 +0.603 }}
: error map: {{val| -0.052 +0.206 -0.039 -0.389 +0.113 -0.190 -0.119 +0.603 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.244{{c}}, ~5/4 = 386.388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2440{{c}}, ~5/4 = 386.3884{{c}}
: error map: {{val| 0.000 +0.289 +0.075 -0.242 +0.293 +0.012 +0.095 +0.860 }}
: error map: {{val| 0.000 +0.289 +0.075 -0.242 +0.293 +0.012 +0.095 +0.860 }}


Optimal ET sequence: TBA
{{Optimal ET sequence|legend=0| 41, 53j, 94, 176, 217, 270, 487, 581j, 1068j }}


Badness (Sintel): 0.758
Badness (Sintel): 0.758
Line 110: Line 110:
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2233/2232
Comma list: 784/783, 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2233/2232


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 23 19 | 0 1 0 -14 23 12 5 -31 -24 | 0 0 1 0 0 -1 1 0 0 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 23 19 | 0 1 0 -14 23 12 5 -31 -24 | 0 0 1 0 0 -1 1 0 0 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.930{{c}}, ~3/2 = 702.208{{c}}, ~5/4 = 386.406{{c}}
* WE: ~2 = 1199.9298{{c}}, ~3/2 = 702.2082{{c}}, ~5/4 = 386.4062{{c}}
: error map: {{val| -0.070 +0.183 -0.048 -0.513 +0.173 -0.225 -0.136 +0.353 +0.633 }}
: error map: {{val| -0.070 +0.183 -0.048 -0.513 +0.173 -0.225 -0.136 +0.353 +0.633 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.250{{c}}, ~5/4 = 386.420{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.2495{{c}}, ~5/4 = 386.4198{{c}}
: error map: {{val| 0.000 +0.295 +0.106 -0.319 +0.421 +0.047 +0.154 +0.688 +0.976 }}
: error map: {{val| 0.000 +0.295 +0.106 -0.319 +0.421 +0.047 +0.154 +0.688 +0.976 }}


Optimal ET sequence: TBA
{{Optimal ET sequence|legend=0| 41, 94, 135, 176, 217, 270, 311, 487, 581jk, 798k, 1068jk }}


Badness (Sintel): 0.793
Badness (Sintel): 0.793
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Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 -41 | 0 1 0 -14 23 12 5 26 | 0 0 1 0 0 -1 1 2 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 -41 | 0 1 0 -14 23 12 5 26 | 0 0 1 0 0 -1 1 2 }}


Optimal tunings:
Optimal tunings:
* WE: ~2 = 1199.983{{c}}, ~3/2 = 702.213{{c}}, ~5/4 = 386.172{{c}}
* WE: ~2 = 1199.9833{{c}}, ~3/2 = 702.2132{{c}}, ~5/4 = 386.1721{{c}}
: error map: {{val| -0.017 +0.242 -0.175 +0.005 -0.246 -0.091 -0.291 +0.495 }}
: error map: {{val| -0.017 +0.242 -0.175 +0.005 -0.246 -0.091 -0.291 +0.495 }}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 702.233{{c}}, ~5/4 = 386.174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2228{{c}}, ~5/4 = 386.1741{{c}}
: error map: {{val| 0.000 +0.268 -0.140 +0.055 -0.193 -0.028 -0.225 +0.564 }}
: error map: {{val| 0.000 +0.268 -0.140 +0.055 -0.193 -0.028 -0.225 +0.564 }}


Optimal ET sequence: TBA
{{Optimal ET sequence|legend=0| 41, 53, 94j, 217j, 258j, 270, 581j, 904, 945, 1215j }}


Badness (Sintel): 0.716
Badness (Sintel): 0.716
Line 146: Line 146:
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079, 2233/2232
Comma list: 1216/1215, 1540/1539, 1625/1624, 1729/1728, 2080/2079, 2233/2232


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -6 -41 -52 | 0 1 0 -14 23 12 5 26 33 | 0 0 1 0 0 -1 1 2 2 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -6 -41 -52 | 0 1 0 -14 23 12 5 26 33 | 0 0 1 0 0 -1 1 2 2 }}


Optimal tunings:
Optimal tunings:
Line 154: Line 154:
: error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}
: error map: {{val| 0.000 +0.264 -0.192 +0.110 -0.284 -0.023 -0.297 +0.358 +0.431 }}


Optimal ET sequence: TBA
{{Optimal ET sequence|legend=0| 41, 53k, 94jk, 217jk, 229j, 270, 311, 581jk, 634, 904, 945, 1215jk }}


Badness (Sintel): 0.796
Badness (Sintel): 0.796


=== Septinoellismic ===
=== Septinoellismic ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 1225/1224, 2080/2079, 2431/2430, 4096/4095
{{Mapping|legend=2| 1 0 0 25 -33 -13 47 | 0 1 0 -14 23 12 -30 | 0 0 1 0 0 -1 2 }}
Optimal tunings:
* WE: ~2 = 1199.9654{{c}}, ~3/2 = 702.2159{{c}}, ~5/4 = 386.2053{{c}}
: error map: {{val| -0.035 +0.226 -0.178 -0.229 -0.006 -0.038 +0.252 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2359{{c}}, ~5/4 = 386.2080{{c}}
: error map: {{val| 0.000 +0.281 -0.106 -0.129 +0.109 +0.096 +0.383 }}
{{Optimal ET sequence|legend=0| 53, 94, 176g, 217, 270, 311, 487, 581 }}
Badness (Sintel): 0.950
==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 164: Line 180:
Comma list: 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2080/2079
Comma list: 1216/1215, 1225/1224, 1540/1539, 1729/1728, 2080/2079


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 47 -6 | 0 1 0 -14 23 12 -30 5 | 0 0 1 0 0 -1 2 1 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 47 -6 | 0 1 0 -14 23 12 -30 5 | 0 0 1 0 0 -1 2 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.970{{c}}, ~3/2 = 702.190{{c}}, ~5/4 = 386.177{{c}}
* WE: ~2 = 1199.9699{{c}}, ~3/2 = 702.2200{{c}}, ~5/4 = 386.2373{{c}}
: error map: {{val|-0.030 +0.235 -0.137 -0.238 +0.044 -0.034 +0.286 -0.206}}
: error map: {{val| -0.030 +0.235 -0.137 -0.238 +0.044 -0.034 +0.286 -0.206 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2307{{c}}, ~5/4 = 386.3245{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2374{{c}}, ~5/4 = 386.2384{{c}}
: error map: {{val|0.000 +0.282 -0.075 -0.150 +0.143 +0.083 +0.399 -0.088}}
: error map: {{val| 0.000 +0.282 -0.075 -0.150 +0.143 +0.083 +0.399 -0.088 }}


{{Optimal ET sequence|legend=0|TBA}}
{{Optimal ET sequence|legend=0| 53, 94, 176g, 217, 270, 311, 487, 581 }}


Badness (Sintel): 0.682
Badness (Sintel): 0.682


=== Septisperasmic ===
=== Septisperasmic ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 2080/2079, 2500/2499, 4096/4095, 19712/19683
{{Mapping|legend=2| 1 0 0 25 -33 -13 -48 | 0 1 0 -14 23 12 27 | 0 0 1 0 0 -1 4 }}
Optimal tunings:
* WE: ~2 = 1199.9797{{c}}, ~3/2 = 702.2164{{c}}, ~5/4 = 386.2286{{c}}
: error map: {{val| -0.020 +0.241 -0.126 -0.080 -0.136 -0.098 +0.068 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2281{{c}}, ~5/4 = 386.2332{{c}}
: error map: {{val| 0.000 +0.273 -0.081 -0.020 -0.071 -0.023 +0.137 }}
{{Optimal ET sequence|legend=0| 258, 270, 311, 581, 1485, 1796, 2066, 2647b }}
Badness (Sintel): 1.76
==== 19-limit ====
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19
Line 182: Line 214:
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2500/2499
Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2500/2499


Subgroup-val mapping: {{mapping| 1 0 0 25 -33 -13 -48 -6 | 0 1 0 -14 23 12 27 5 | 0 0 1 0 0 -1 4 1 }}
{{Mapping|legend=2| 1 0 0 25 -33 -13 -48 -6 | 0 1 0 -14 23 12 27 5 | 0 0 1 0 0 -1 4 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.986{{c}}, ~3/2 = 702.206{{c}}, ~5/4 = 386.207{{c}}
* WE: ~2 = 1199.9861{{c}}, ~3/2 = 702.2198{{c}}, ~5/4 = 386.2353{{c}}
: error map: {{val|-0.014 +0.251 -0.106 -0.056 -0.123 -0.084 +0.101 -0.193}}
: error map: {{val| -0.014 +0.251 -0.106 -0.056 -0.123 -0.084 +0.101 -0.193 }}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.228{{c}}, ~5/4 = 386.238{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.2278{{c}}, ~5/4 = 386.2383{{c}}
: error map: {{val|0.000 +0.273 -0.075 -0.015 -0.078 -0.032 +0.148 -0.136}}
: error map: {{val| 0.000 +0.273 -0.075 -0.015 -0.078 -0.032 +0.148 -0.136 }}


{{Optimal ET sequence|legend=0|TBA}}
{{Optimal ET sequence|legend=0| 258, 270, 311, 581, 1485, 1796, 2066 }}


Badness (Sintel): 1.096
Badness (Sintel): 1.10


== Androschismic ==
== Androschismic ==
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Comma list: 2080/2079, 10648/10647, 43904/43875
Comma list: 2080/2079, 10648/10647, 43904/43875


Mapping: {{mapping| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}
{{Mapping|legend=0| 1 0 0 25 62 82 | 0 1 0 -14 -34 -43 | 0 0 1 0 -2 -3 }}


Optimal tunings:  
Optimal tunings:  
Line 231: Line 263:
Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647
Comma list: 1216/1215, 2080/2079, 3136/3135, 10648/10647


Mapping: {{mapping| 1 0 0 25 62 82 -6 | 0 1 0 -14 -34 -43 5 | 0 0 1 0 -2 -3 1 }}
{{Mapping|legend=0| 1 0 0 25 62 82 -6 | 0 1 0 -14 -34 -43 5 | 0 0 1 0 -2 -3 1 }}


Optimal tunings:  
Optimal tunings:  
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